Olympiad problems

2013 Saudi Arabia BMO TST, 2

The base-$7$ representation of number $n$ is $\overline{abc}_{(7)}$, and the base-$9$ representation of number $n$ is $\overline{cba}_{(9)}$. What is the decimal (base-$10$) representation of $n$?

2023 Romanian Master of Mathematics Shortlist, N2

Tags: number theory , decimal representation , algebra , polynomial

For every non-negative integer $k$ let $S(k)$ denote the sum of decimal digits of $k$. Let $P(x)$ and $Q(x)$ be polynomials with non-negative integer coecients such that $S(P(n)) = S(Q(n))$ for all non-negative integers $n$. Prove that there exists an integer $t$ such that $P(x) - 10^tQ(x)$ is a constant polynomial.

1975 IMO Shortlist, 5

Tags: algebra , digit , series summation , inequality , decimal representation

Let $M$ be the set of all positive integers that do not contain the digit $9$ (base $10$). If $x_1, \ldots , x_n$ are arbitrary but distinct elements in $M$, prove that \[\sum_{j=1}^n \frac{1}{x_j} < 80 .\]

2004 IMO, 6

Tags: number theory , decimal representation , modular arithmetic , induction

We call a positive integer [i]alternating[/i] if every two consecutive digits in its decimal representation are of different parity. Find all positive integers $n$ such that $n$ has a multiple which is alternating.

1990 IMO Shortlist, 20

Tags: pigeonhole principle , divisibility , digit , decimal representation , multiple , number theory

Prove that every integer $ k$ greater than 1 has a multiple that is less than $ k^4$ and can be written in the decimal system with at most four different digits.

1960 IMO Shortlist, 1

Tags: divisibility , decimal representation , algebra , number theory

Determine all three-digit numbers $N$ having the property that $N$ is divisible by 11, and $\dfrac{N}{11}$ is equal to the sum of the squares of the digits of $N$.

2011 Junior Balkan Team Selection Tests - Romania, 1

Tags: decimal representation , digit , number theory

Call a positive integer [i]balanced [/i] if the number of its distinct prime factors is equal to the number of its digits in the decimal representation; for example, the number $385 = 5 \cdot 7 \cdot 11$ is balanced, while $275 = 5^2 \cdot 11$ is not. Prove that there exist only a finite number of balanced numbers.

2018 Federal Competition For Advanced Students, P1, 3

Tags: decimal representation , digit , combinatorics , game , number theory

Alice and Bob determine a number with $2018$ digits in the decimal system by choosing digits from left to right. Alice starts and then they each choose a digit in turn. They have to observe the rule that each digit must differ from the previously chosen digit modulo $3$. Since Bob will make the last move, he bets that he can make sure that the final number is divisible by $3$. Can Alice avoid that? [i](Proposed by Richard Henner)[/i]

2003 Junior Balkan Team Selection Tests - Romania, 2

Tags: odd , decimal representation , power , number theory

Let $a$ be a positive integer such that the number $a^n$ has an odd number of digits in the decimal representation for all $n > 0$. Prove that the number $a$ is an even power of $10$.

2013 IMO Shortlist, N4

Tags: perfect square , decimal representation , number theory

Determine whether there exists an infinite sequence of nonzero digits $a_1 , a_2 , a_3 , \cdots $ and a positive integer $N$ such that for every integer $k > N$, the number $\overline{a_k a_{k-1}\cdots a_1 }$ is a perfect square.

1975 IMO, 4

Tags: divisibility , digit , decimal representation , digit sum , modular arithmetic

When $4444^{4444}$ is written in decimal notation, the sum of its digits is $ A.$ Let $B$ be the sum of the digits of $A.$ Find the sum of the digits of $ B.$ ($A$ and $B$ are written in decimal notation.)

1984 Spain Mathematical Olympiad, 7

Tags: digit , decimal representation , number theory

Consider the natural numbers written in the decimal system. (a) Find the smallest number which decreases five times when its first digit is erased. Which form do all numbers with this property have? (b) Prove that there is no number that decreases $12$ times when its first digit is erased. (c) Find the necessary and sufficient condition on $k$ for the existence of a natural number which is divided by $k$ when its first digit is erased.

2003 IMO Shortlist, 4

Tags: decimal representation , perfect square , number theory , modular arithmetic

Let $ b$ be an integer greater than $ 5$. For each positive integer $ n$, consider the number \[ x_n = \underbrace{11\cdots1}_{n \minus{} 1}\underbrace{22\cdots2}_{n}5, \] written in base $ b$. Prove that the following condition holds if and only if $ b \equal{} 10$: [i]there exists a positive integer $ M$ such that for any integer $ n$ greater than $ M$, the number $ x_n$ is a perfect square.[/i] [i]Proposed by Laurentiu Panaitopol, Romania[/i]

1966 IMO Shortlist, 12

Tags: equation , decimal representation , number theory

Find digits $x, y, z$ such that the equality \[\sqrt{\underbrace{\overline{xx\cdots x}}_{2n \text{ times}}-\underbrace{\overline{yy\cdots y}}_{n \text{ times}}}=\underbrace{\overline{zz\cdots z}}_{n \text{ times}}\] holds for at least two values of $n \in \mathbb N$, and in that case find all $n$ for which this equality is true.

2007 Germany Team Selection Test, 3

Tags: rational , decimal representation , number theory

For $ x \in (0, 1)$ let $ y \in (0, 1)$ be the number whose $ n$-th digit after the decimal point is the $ 2^{n}$-th digit after the decimal point of $ x$. Show that if $ x$ is rational then so is $ y$. [i]Proposed by J.P. Grossman, Canada[/i]

1980 IMO, 3

Tags: decimal representation , digit , modular arithmetic , combinatorics

Find the digits left and right of the decimal point in the decimal form of the number \[ (\sqrt{2} + \sqrt{3})^{1980}. \]

1969 IMO Shortlist, 40

Tags: decimal representation , counting , digit , combinatorics

$(MON 1)$ Find the number of five-digit numbers with the following properties: there are two pairs of digits such that digits from each pair are equal and are next to each other, digits from different pairs are different, and the remaining digit (which does not belong to any of the pairs) is different from the other digits.

1981 Tournament Of Towns, (013) 3

Tags: decimal representation , sum , number theory

Prove that every real positive number may be represented as a sum of nine numbers whose decimal representation consists of the digits $0$ and $7$. (E Turkevich)

2014 Belarus Team Selection Test, 3

Tags: perfect square , decimal representation , number theory

Determine whether there exists an infinite sequence of nonzero digits $a_1 , a_2 , a_3 , \cdots $ and a positive integer $N$ such that for every integer $k > N$, the number $\overline{a_k a_{k-1}\cdots a_1 }$ is a perfect square.

2015 Balkan MO Shortlist, N7

Tags: decimal representation , digit , number theory

Positive integer $m$ shall be called [i]anagram [/i] of positive $n$ if every digit $a$ appears as many times in the decimal representation of $m$ as it appears in the decimal representation of $n$ also. Is it possible to find $4$ different positive integers such that each of the four to be [i]anagram [/i] of the sum of the other $3$? (Bulgaria)

2001 Switzerland Team Selection Test, 5

Tags: number theory , digit , sum , decimal representation , inequalities

Let $a_1 < a_2 < ... < a_n$ be a sequence of natural numbers such that for $i < j$ the decimal representation of $a_i$ does not occur as the leftmost digits of the decimal representation of $a_j$ . (For example, $137$ and $13729$ cannot both occur in the sequence.) Prove that $\sum_{i=1}^n \frac{1}{a_i} \le 1+\frac12 +\frac13 +...+\frac19$ .

1967 IMO Longlists, 1

Tags: decimal representation , perfect cube , algebra , number theory

Prove that all numbers of the sequence \[ \frac{107811}{3}, \quad \frac{110778111}{3}, \frac{111077781111}{3}, \quad \ldots \] are exact cubes.

1986 China Team Selection Test, 3

Tags: decimal representation , number theoretic functions , digit , algebra

Given a positive integer $A$ written in decimal expansion: $(a_{n},a_{n-1}, \ldots, a_{0})$ and let $f(A)$ denote $\sum^{n}_{k=0} 2^{n-k}\cdot a_k$. Define $A_1=f(A), A_2=f(A_1)$. Prove that: [b]I.[/b] There exists positive integer $k$ for which $A_{k+1}=A_k$. [b]II.[/b] Find such $A_k$ for $19^{86}.$

1983 IMO Shortlist, 24

Tags: decimal representation , digit , periodic sequence , non-periodical functions , number theory

Let $d_n$ be the last nonzero digit of the decimal representation of $n!$. Prove that $d_n$ is aperiodic; that is, there do not exist $T$ and $n_0$ such that for all $n \geq n_0, d_{n+T} = d_n.$

2000 Belarus Team Selection Test, 5.2

Tags: sum of digits , decimal representation , number theory

Let $n,k$ be positive integers such that n is not divisible by 3 and $k \geq n$. Prove that there exists a positive integer $m$ which is divisible by $n$ and the sum of its digits in decimal representation is $k$.